product of injective modules is injective

Proof. Let B be an arbitrary R-module, A⊆B a submodule and f:A→Q a homomorphism. It is enough to show that f can be extended to B. For i∈I denote by πi:Q→Qi the projection. Since Qi is injective for any i, then the homomorphism πi∘f:A→Qi can be extended to fi′:B→Qi. Then we have

f′:B→Q;

f′⁢(b)=(fi′⁢(b))i∈I.

It is easy to check, that if a∈A, then f′⁢(a)=f⁢(a), so f′ is an extension of f. Thus Q is injective. □

Remark. Unfortunetly direct sum of injective modules need not be injective. Indeed, there is a theorem which states that direct sums of injective modules are injective if and only if ring R is Noetherian. Note that the proof presented above cannot be used for direct sums, because f′⁢(b) need not be an element of the direct sum, more precisely, it is possible that fi′⁢(b)≠0 for infinetly many i∈I. Nevertheless products are always injective.